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Properties

Label 1425.2.c.j.799.1
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1425,2,Mod(799,1425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,-4,0,0,-4,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.j.799.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -3.41421i q^{7} +4.41421i q^{8} -1.00000 q^{9} -1.41421 q^{11} +3.82843i q^{12} +2.58579i q^{13} -8.24264 q^{14} +3.00000 q^{16} +6.82843i q^{17} +2.41421i q^{18} -1.00000 q^{19} -3.41421 q^{21} +3.41421i q^{22} -3.65685i q^{23} +4.41421 q^{24} +6.24264 q^{26} +1.00000i q^{27} +13.0711i q^{28} -5.07107 q^{29} -10.4853 q^{31} +1.58579i q^{32} +1.41421i q^{33} +16.4853 q^{34} +3.82843 q^{36} +3.07107i q^{37} +2.41421i q^{38} +2.58579 q^{39} -4.58579 q^{41} +8.24264i q^{42} +3.41421i q^{43} +5.41421 q^{44} -8.82843 q^{46} -11.6569i q^{47} -3.00000i q^{48} -4.65685 q^{49} +6.82843 q^{51} -9.89949i q^{52} +4.00000i q^{53} +2.41421 q^{54} +15.0711 q^{56} +1.00000i q^{57} +12.2426i q^{58} +8.48528 q^{59} -5.65685 q^{61} +25.3137i q^{62} +3.41421i q^{63} +9.82843 q^{64} +3.41421 q^{66} -12.0000i q^{67} -26.1421i q^{68} -3.65685 q^{69} +12.4853 q^{71} -4.41421i q^{72} -2.00000i q^{73} +7.41421 q^{74} +3.82843 q^{76} +4.82843i q^{77} -6.24264i q^{78} -11.3137 q^{79} +1.00000 q^{81} +11.0711i q^{82} +6.48528i q^{83} +13.0711 q^{84} +8.24264 q^{86} +5.07107i q^{87} -6.24264i q^{88} +14.7279 q^{89} +8.82843 q^{91} +14.0000i q^{92} +10.4853i q^{93} -28.1421 q^{94} +1.58579 q^{96} -4.24264i q^{97} +11.2426i q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 16 q^{14} + 12 q^{16} - 4 q^{19} - 8 q^{21} + 12 q^{24} + 8 q^{26} + 8 q^{29} - 8 q^{31} + 32 q^{34} + 4 q^{36} + 16 q^{39} - 24 q^{41} + 16 q^{44} - 24 q^{46} + 4 q^{49}+ \cdots + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.41421i − 1.70711i −0.521005 − 0.853553i \(-0.674443\pi\)
0.521005 − 0.853553i \(-0.325557\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) − 3.41421i − 1.29045i −0.763992 − 0.645226i \(-0.776763\pi\)
0.763992 − 0.645226i \(-0.223237\pi\)
\(8\) 4.41421i 1.56066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 − 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 3.82843i 1.10517i
\(13\) 2.58579i 0.717168i 0.933497 + 0.358584i \(0.116740\pi\)
−0.933497 + 0.358584i \(0.883260\pi\)
\(14\) −8.24264 −2.20294
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843i 1.65614i 0.560627 + 0.828068i \(0.310560\pi\)
−0.560627 + 0.828068i \(0.689440\pi\)
\(18\) 2.41421i 0.569036i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.41421 −0.745042
\(22\) 3.41421i 0.727913i
\(23\) − 3.65685i − 0.762507i −0.924471 − 0.381253i \(-0.875493\pi\)
0.924471 − 0.381253i \(-0.124507\pi\)
\(24\) 4.41421 0.901048
\(25\) 0 0
\(26\) 6.24264 1.22428
\(27\) 1.00000i 0.192450i
\(28\) 13.0711i 2.47020i
\(29\) −5.07107 −0.941674 −0.470837 − 0.882220i \(-0.656048\pi\)
−0.470837 + 0.882220i \(0.656048\pi\)
\(30\) 0 0
\(31\) −10.4853 −1.88321 −0.941606 − 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) 1.58579i 0.280330i
\(33\) 1.41421i 0.246183i
\(34\) 16.4853 2.82720
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 3.07107i 0.504880i 0.967612 + 0.252440i \(0.0812331\pi\)
−0.967612 + 0.252440i \(0.918767\pi\)
\(38\) 2.41421i 0.391637i
\(39\) 2.58579 0.414057
\(40\) 0 0
\(41\) −4.58579 −0.716180 −0.358090 − 0.933687i \(-0.616572\pi\)
−0.358090 + 0.933687i \(0.616572\pi\)
\(42\) 8.24264i 1.27187i
\(43\) 3.41421i 0.520663i 0.965519 + 0.260331i \(0.0838318\pi\)
−0.965519 + 0.260331i \(0.916168\pi\)
\(44\) 5.41421 0.816223
\(45\) 0 0
\(46\) −8.82843 −1.30168
\(47\) − 11.6569i − 1.70033i −0.526519 − 0.850163i \(-0.676503\pi\)
0.526519 − 0.850163i \(-0.323497\pi\)
\(48\) − 3.00000i − 0.433013i
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) 6.82843 0.956171
\(52\) − 9.89949i − 1.37281i
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 2.41421 0.328533
\(55\) 0 0
\(56\) 15.0711 2.01396
\(57\) 1.00000i 0.132453i
\(58\) 12.2426i 1.60754i
\(59\) 8.48528 1.10469 0.552345 − 0.833616i \(-0.313733\pi\)
0.552345 + 0.833616i \(0.313733\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 − 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 25.3137i 3.21484i
\(63\) 3.41421i 0.430150i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) 3.41421 0.420261
\(67\) − 12.0000i − 1.46603i −0.680211 − 0.733017i \(-0.738112\pi\)
0.680211 − 0.733017i \(-0.261888\pi\)
\(68\) − 26.1421i − 3.17020i
\(69\) −3.65685 −0.440234
\(70\) 0 0
\(71\) 12.4853 1.48173 0.740865 − 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) − 4.41421i − 0.520220i
\(73\) − 2.00000i − 0.234082i −0.993127 − 0.117041i \(-0.962659\pi\)
0.993127 − 0.117041i \(-0.0373409\pi\)
\(74\) 7.41421 0.861885
\(75\) 0 0
\(76\) 3.82843 0.439151
\(77\) 4.82843i 0.550250i
\(78\) − 6.24264i − 0.706840i
\(79\) −11.3137 −1.27289 −0.636446 − 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 11.0711i 1.22259i
\(83\) 6.48528i 0.711852i 0.934514 + 0.355926i \(0.115835\pi\)
−0.934514 + 0.355926i \(0.884165\pi\)
\(84\) 13.0711 1.42617
\(85\) 0 0
\(86\) 8.24264 0.888827
\(87\) 5.07107i 0.543676i
\(88\) − 6.24264i − 0.665468i
\(89\) 14.7279 1.56116 0.780578 − 0.625058i \(-0.214925\pi\)
0.780578 + 0.625058i \(0.214925\pi\)
\(90\) 0 0
\(91\) 8.82843 0.925471
\(92\) 14.0000i 1.45960i
\(93\) 10.4853i 1.08727i
\(94\) −28.1421 −2.90264
\(95\) 0 0
\(96\) 1.58579 0.161849
\(97\) − 4.24264i − 0.430775i −0.976529 − 0.215387i \(-0.930899\pi\)
0.976529 − 0.215387i \(-0.0691014\pi\)
\(98\) 11.2426i 1.13568i
\(99\) 1.41421 0.142134
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1425.2.c.j.799.1 4
5.2 odd 4 285.2.a.f.1.2 ✓ 2
5.3 odd 4 1425.2.a.l.1.1 2
5.4 even 2 inner 1425.2.c.j.799.4 4
15.2 even 4 855.2.a.e.1.1 2
15.8 even 4 4275.2.a.x.1.2 2
20.7 even 4 4560.2.a.bj.1.1 2
95.37 even 4 5415.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.2 ✓ 2 5.2 odd 4
855.2.a.e.1.1 2 15.2 even 4
1425.2.a.l.1.1 2 5.3 odd 4
1425.2.c.j.799.1 4 1.1 even 1 trivial
1425.2.c.j.799.4 4 5.4 even 2 inner
4275.2.a.x.1.2 2 15.8 even 4
4560.2.a.bj.1.1 2 20.7 even 4
5415.2.a.p.1.1 2 95.37 even 4